Beam halo and growth of beam emittance are important issues for high-intensity linear accelerators. Beam-dynamic predictions of weak beam tails are usually not reliable due to complexity of the non-linear effects leading to halo formation. Therefore, development of a simple method for beam halo diagnostics is highly desirable. The first testing of CR-39 solid-state nuclear track dosimeters for beam halo measurement were performed at the SARAF phase I accelerator with a few MeV proton beams. Beam pulses of 90 nA peak intensity of shortest possible duration (15 ns) were used for direct irradiation of standard CR-39 personal dosimetry tags. Other irradiations were done with beam pulses of 200 ns duration and of 1 mA peak intensity. Specially prepared large area CR-39 plates with central hole for the beam core transport were used in these tests. Weak beam structures were clearly observed in the both types of irradiation. The tests showed feasibility of beam halo measurements down to resolution level of a single proton. The optimum CR-39 etching conditions were established. The advantages and drawback of the method are discussed.